Math teachers: here's an end-of-term puzzle for your class!

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The Four Fours Puzzle by ChatGPT (v. 5.5 thinking)

As the summer term came to an end, Nigel, in his teaching days, found that the puzzle below got a maths class interested very quickly, especially when tackled collaboratively in pairs, groups or competing teams.

The problem is simple to state. Can you make every whole number from 1 to 100 using exactly four 4s and standard arithmetic operations? 

For example, 2 can be made as:

4/4 + 4/4 = 2

Every expression must contain exactly four 4s. The operations may include addition, subtraction, multiplication, division, brackets, square roots, powers, factorials, decimal points, recurring decimals and joining two 4s together to make 44.

In the solutions below, 4! means 4 factorial, so 4! = 24. A dot above a 4 means that the 4 recurs: for example, .4̇ means 0.4444..., and .44̇ also means 0.4444..., but uses two written 4s.

1	44/44
2	4 × (4/(4 + 4))
3	(4 + 4 + 4)/4
4	4 + 4 × (4 − 4)
5	(4 + 4 × 4)/4
6	4 + (4 + 4)/4
7	44/4 − 4
8	4 + 4 + 4 − 4
9	4 + 4 + 4/4
10	(44 − 4)/4
11	44/√(4 × 4)
12	(4 + 44)/4
13	√4 + 44/4
14	4 + 4 + 4 + √4
15	4 + 44/4
16	4 + 4 + 4 + 4
17	4 × 4 + 4/4
18	44/√4 − 4
19	4! − 4/4 − 4
20	4 × (4 + 4/4)
21	4! + 4/4 − 4
22	√4 × 44/4
23	(4 × 4! − 4)/4
24	4 + 4 + 4 × 4
25	(4 + 4 × 4!)/4
26	4 + 44/√4
27	4 + 4! − 4/4
28	44 − 4 × 4
29	4 + 4! + 4/4
30	4 × (4 + 4) − √4
31	4! + (4 + 4!)/4
32	4 × 4 + 4 × 4
33	4 + 4! + √4/.4
34	√4 + 4 × (4 + 4)
35	4! + 44/4
36	44 − 4 − 4
37	4! + (√4 + 4!)/√4
38	44 − √4 − 4
39	44 − √4/.4
40	44 − √(4 × 4)
41	(.4 + 4 × 4)/.4
42	√4 + 44 − 4
43	44 − 4/4
44	4 + 44 − 4
45	44 + 4/4
46	4 − (√4 − 44)
47	√4 × 4! − 4/4
48	4 × (4 + 4 + 4)
49	4/4 + √4 × 4!
50	4 + √4 + 44
51	((4! − √4)/.4) − 4
52	4 + 4 + 44
53	44 + 4/.4̇
54	44 + 4/.4
55	(44/√4)/.4
56	4 × (4 × 4 − √4)
57	√4 − (√4 − 4!)/.4
58	(44 − 4!)/4
59	4!/.4 − 4/4
60	44 + 4 × 4
61	4/4 + 4!/.4
62	4 × 4 × 4 − √4
63	(44 − 4)/4
64	(4 + 4) × (4 + 4)
65	(4 + 44)/4
66	√4 + 4 × 4 × 4
67	√4 + (√4 + 4!)/.4
68	4 + 4 × 4 × 4
69	4 + (√4 + 4!)/.4
70	√4 + 4! + 44
71	(4! + 4.4)/.4
72	4 + 4! + 44
73	(√(.4̇) + √4 × 4!)/√(.4̇)
74	4 + (4 + 4!)/.4
75	(4 + √4 + 4!)/.4
76	4 × (4! − 4) − 4
77	(√(4/.4̇))4 − 4
78	4 × (4! − 4) − √4
79	4! − (√4 − 4!)/.4
80	4 × (4 + 4 × 4)
81	(4 − 4/4)4
82	√4 − 4 × (4 − 4!)
83	4! − (.4 − 4!)/.4
84	√4 × 44 − 4
85	(4! + 4/.4)/.4
86	√4 × 44 − √4
87	4 × 4! − 4/.4̇
88	44 + 44
89	4! + (√4 + 4!)/.4
90	√4 + √4 × 44
91	4 × 4! − √4/.4
92	4 + √4 × 44
93	4 × 4! − √(4/.4̇)
94	√4 + 4 × 4! − 4
95	4 × 4! − 4/4
96	√4 × (4 + 44)
97	4/4 + 4 × 4!
98	4 − (√4 − 4 × 4!)
99	4.4/(.4̇ − .4)
100	4 × (4! + 4/4)

Some numbers are easy to construct, others bafflingly hard - tell them not to try working through 1-100 in order, go for the low-hanging fruit first.

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Yes, you should worry; it is weird!

When Reality Changes but No One Tells the Children

It's a regular experience in maths and science education: you're introduced to an idea that is, quite obviously, bizarre. The teacher, however, presents it with calm indifference, as if this conceptual landslide were no more than a change in bus timetable. You ask a question. The room shifts. They mutter something about "just notation" and move on. You’re left wondering: did reality just warp, or am I the only one paying attention?

1. Imaginary Numbers

Once denounced as a monstrous fiction - what kind of number squares to –1? - the imaginary unit i now appears in A-level maths without apology. It lives on a plane no one can see, invented solely to make polynomials behave and and casually invokes the spooky masterpiece e^iπ = –1.

Is any of this stuff real? (Pun intended.)

2. Infinity

Once the playground of mystics, now a fixture in every calculus class. We sum to infinity, we integrate to infinity; just don’t ask what infinity actually is. The teacher will wave it off as “just notation” and quietly hope you don’t bring up Zeno.

Bishop Berkeley wouldn’t have let that slide.

3. Transfinite Numbers

Cantor proved some infinities are bigger than others. He was called a heretic and died in a sanitorium. Today, ℵ₀ and ℵ₁ are chalked up like counting numbers. Whether they actually exist is left to metaphysics - or madness.

We now play with infinite sets like Lego bricks. It's best not to ask where the ceiling is.

4. Numbers as Real Entities

What is the number 2? You’ve never seen it. It’s nowhere in space or time. Yet it's treated as if it sits beside your pencil case. Ask where numbers live, and your teacher looks faintly alarmed and has no reply.

You’ve just tripped over Platonism. Don’t expect anyone to mention it.

5. Action at a Distance

Newton hated the idea that gravity could reach out across the void. So did Berkeley. Yet GCSE physics has the Sun tugging Earth like a yo-yo on a string. Fields show up later as a fix, but how exactly does a “field” live in empty space which is... empty?

This was once considered an ontological scandal. Now it’s a diagram with arrows.

6. Energy as a Property

We're taught energy is “stored” in objects like jam in a doughnut. But energy depends on your frame of reference. It’s not intrinsic, it’s relational. Try explaining that to someone sliding boxes down ramps.

You'll have to wait for graduate school before they sort that one out for you.

7. The Reality of Atoms

Atoms were controversial well into the 20th century. Today we draw them like tiny solar systems and teach them to children. That model’s false, but don’t worry: the reality is fuzzier, deeper, and no one understands it anyway.

Also: we have no idea what an electron actually is. Just smile and say “cloud.”

8. Orbitals as Clouds

“Here’s where the electron probably is,” says the textbook, handing you a pastel blob. Strictly speaking, the blob shows a probability density in 3D space, even though the wavefunction itself lives in an abstract mathematical configuration space (Hilbert space). The distinction is usually swept under the rug... until you ask about two electrons at once.

So what does that mean? Never mind; just colour it in neatly.

9. Evolution by Natural Selection

Once blasphemous, now doctrine. But the theory’s metaphysics - brutality, randomness, directionlessness - rarely get aired. “Organisms adapt” is euphemistic code for “bad stuff happens and the winners reproduce.”

Also: don’t ask about recent human evolution. That’s quietly absent from the syllabus.

10. The Species Concept

Biology likes boxes: genus, species, subspecies. The genome laughs: no species there. Hybridisation, gene flow, ring species* - they all blur the lines. Still, you’ll be tested on Linnaean clarity.

“It’s complicated,” says the teacher. Then they'll mark you as wrong.

11. The Arrow of Time

Entropy increases. That’s that. But the laws of physics don’t care about direction. So why does time seem to flow one way? “Because, for some unknown reason, the Big Bang had extremely low entropy,” the teacher 'explains'. 

'Well, glad that's been sorted out,' you say to yourself.

12. Simultaneity in Relativity

One person’s 'now' is another’s 'not yet' and someone else's 'already happened'. Simultaneity is relative. Einstein took years to accept it. The equations and diagrams purport to make it clear, but no one can visualise Minkowski spacetime. And if the future is as real as the past and present - wow, that's pretty mind-blowing!

Welcome to spacetime as it really is.

13. Wave–Particle Duality

Light is a wave. No, it’s a particle. Actually it’s both. Or neither. It depends who’s asking and how. This was once a crisis; now it’s a footnote. The cat is alive and dead. No one blinks.

Don’t think: just calculate.

14. Energy Non-Conservation in Cosmology

Energy is conserved, unless spacetime is curved and expanding. In which case, not so much. Turns out those mysterious “conservation laws” actually arise from the time and space symmetries of the laws of physics, and if you bend the stage, the rules shift.

Who's ever heard of Emmy Noether^**, anyway?

15. Calculus and Infinitesimals

We teach limits to sanitise calculus, banishing infinitesimals as ghostly nonsense. Then we write dx and treat it like a tiny thing. Don't ask.

Later, Abraham Robinson sneaks infinitesimals back in; now rigorously defined, they're legit again.

Students still don’t know what dx is. Nor, frankly, do most teachers.

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* A ring species forms when a population spreads out geographically in a ring-like fashion around a geographic barrier (like a mountain or a desert). Along the way, neighbouring populations can interbreed with each other, gradually changing as they go. But when the two ends of the ring meet again, the terminal populations are so different that they can no longer interbreed.This entirely subverts the transitive definition of 'species'.

** In general relativity, energy conservation fails because the time-translation symmetry needed for Noether’s theorem doesn’t exist globally (eg if the metric is time-varying as in the FRW case). Emmy Noether, who discovered the deep link between symmetry and conservation laws, is mostly unknown to students, despite having written one of the most important theorems in physics. There is no justice.

How Augustine Met His Wife

 

The 1980s family which might have inspired this tale

It was nearing the conclusion of the summer term when Mr. Augustine, a young man of gentle disposition and a mind for numbers, first encountered the object of his affections. Daily, he would ferry Miss Susan, a governess entrusted with the education of history, back from their place of employment. Despite their close quarters within the confines of his gig, a state of affairs lasting a full ninety minutes, their connection remained curiously platonic. Miss Susan, though pleasant enough, failed to stir even the faintest ripple in the well of his emotions. Indeed, it was a peculiar sensation, this lack of sparkle in the presence of the fairer sex.

One fateful day, however, Miss Susan presented Mr. Augustine with a surprise. Upon entering her humble abode, his gaze fell upon a vision unlike any he had encountered before. Reclining upon the sofa, as graceful as a Grecian statue come to life, was a young woman whose beauty rivaled the goddess Venus herself.

Mr. Augustine, weary of the relentless demands of his current profession, had long contemplated a change of course. The drudgery of instructing the lower echelons in mathematics, a duty bestowed upon him solely by virtue of his junior status, had thoroughly dampened his spirits. The very institution, once envisioned as a place of learning, now resembled a prison, its teachers akin to warders burdened by the constant barrage of complaints. By the year's end, the staffroom had become a haze of cigarette smoke, a refuge for weary souls coughing out their frustrations.

No longer could Mr. Augustine endure such a fate. A well-deserved vacation awaited him, followed by a path towards retraining as a software engineer, a profession promising a far more substantial remuneration. Only now, with this newfound resolve, did his life seem poised to truly begin.

The slumbering beauty was roused by their arrival. Miss Susan, ever the gracious hostess, had proposed a celebratory curry to mark Mr. Augustine's impending departure. Truth be told, his options for that evening were rather limited.

Miss Susan’s friend, while undeniably attractive, possessed a personality that leaned towards the serious, even combative. A simple remark from Mr. Augustine, expressing his relief at leaving the teaching profession, would be met with a sharp retort.

"Are you any good at mathematics, Mr. Augustine?" she might inquire, a note of challenge in her voice.

"Indeed, I daresay I possess a certain aptitude," he would reply, a hint of pride colouring his tone. "Perhaps the most adept within the department," he might add - with the silent acknowledgement that such a distinction held little weight in their current circumstances.

"Then you desert these young minds in favour of personal comfort?" she would counter, her brows furrowed in disapproval.

"My own superior," he would respond, launching into a detailed account, "was recently forced to retire due to a nervous ailment brought on by the relentless stress. I fear a similar fate awaits me if I do not make a change."

In truth, Mr. Augustine could already feel the telltale signs – a creeping detachment from reality, a dulling of his senses.

Miss Susan’s beautiful friend remained unconvinced. "So many abandon their posts," she would declare, her voice laced with indignation. "And there was that recent pay increase, one must not forget!"

While Mr. Augustine suspected such statistics might well support his side of the argument, their debate continued well into their meal, their verbal sparring as fiery as the curry itself.

It was perhaps inevitable that, amidst such spirited exchanges, an invitation should be extended. To his surprise, it was readily accepted. As he would later discover, Miss Susan's friend's arguments often served as a curious prelude to affection, a means of clearing the air before something more might blossom.

Their connection, though not one struck by lightning, was nonetheless genuine. A comfortable companionship formed the bedrock of their relationship. Beneath the surface of their playful jabs and intellectual jousting, a subtle attraction simmered. In their absences, each felt adrift, incomplete, as if searching for a missing piece.

Alas, a single, colossal argument proved their undoing, leading to a bitter parting of ways. Mr. Augustine, with a heavy heart, recorded the unfortunate event in his personal diary: "So much for that!"

Three long months passed before Miss Susan’s friend recognized the gravity of her error. A single day was all it took to rectify the situation. Given the confines of their small, bohemian community, arranging a chance encounter was a simple feat. And Mr. Augustine, never one to resist the charms of a spirited young lady, readily responded to her renewed interest.

Their tale, alas, cannot be neatly concluded with a flourish of "happily ever after." He, a creature of the intellect, found solace in the realm of ideas, thriving in his newfound profession. She, on the other hand, possessed a spirit as wild as the untamed countryside, yearning for constant motion and the thrill of encountering the unknown.

This disparity, naturally, led to clashes. Tempers flared, grievances were nursed, and reconciliations were laborious affairs. Yet, a curious truth remained: when separated for even a short while, their lives seemed drained of vibrancy, a dull ache settling where connection had once thrummed. The mere presence of one another rekindled a spark, igniting reunions with a sweetness that only absence could intensify.

With the passage of time, they learned to navigate their differences with greater grace. Theirs was a dance, a constant ebb and flow between intellectual pursuits and adventurous expeditions. Perhaps not a conventional happily ever after, but a harmony born of acceptance and a deep-seated affection that defied easy categorization.

One sunny afternoon, as their children embarked on the tempestuous journey of adolescence, Augustine turned to them with a twinkle in his eye. "As for the finer details of our courtship," he declared, "that, my dears, is a tale best left to your mother. Good luck prying it from her!"

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© Adam Carlton: with thanks to Gemini Pro for moving the story 210 years backwards in time and hundreds of miles northwest to Chawton, GU34 1SD.

Scale discrepancies

In the summer of 1977 I decided to leave teaching. I had spent a couple of years at Ormonde High School (now Maghull High School), an ex-secondary modern in Maghull, Liverpool teaching maths to the 11-16 year olds. I think it would be fair to say that my students were at the lower end of the ability range. I was burnt out and exhausted.

At that time computers were entering practical use in industry and there was an increasing need for programmers. The Government set up the so-called TOPS courses (Training Opportunities Scheme), which subsidised people changing career. I passed the tests and started a 20 week course to learn COBOL programming with KBS Computer Services in Dale Street, Liverpool. The class was told the course would be really intensive, seven day working , evenings thrown in, partners expected to be abandoned for the duration.

I was in competition with a guy named Paul. He was intuitive, writing code festooned with GOTO statements. I was more organised, knew that GOTO was considered harmful and taught myself structured programming. We were both taken on by KBS at the end of the course.

One module was IBM S/360 assembly programming. I recall working on a program with maybe a hundred or so instructions - load this register, add packed, and so on. I thought: 'It's taking me an hour or more to get this code right, yet when it executes it will take 100 microseconds'.

I found that scale discrepancy rather disquieting. It's true of every program but in assembler it's somehow more visceral.

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We ran our programs on punch cards through the IBM 360 owned by the Merseyside Docks and Harbour Company. The machine was on the top floor of the Liver Building at Liverpool Pier Head, and we'd go in the evening when the machine was not in regular use.

Access was easy. It wouldn't work today. I once took Clare (who I was 'going out with' at the time) to see the unimpressive sight of those big blue cabinets and the dinner plate sized removable disk drives (20 Megabytes). She does not recall an epiphany.

The United Socialist States of Europe

In the mid-1970s when I was trying to escape from teaching, I applied for an executive officer job with the Civil Service. I took the tests and attended an interview where I was asked, "What do you think about Britain being in the Common Market?".

I was conflicted: I was still a revolutionary socialist so I knew the party line was: 'No to the Europe of the capitalists, for a United Socialist States of Europe!'.

On the other hand, I was a disillusioned leftist and inclined to doubt that insurrection was likely to engulf Europe any time soon. In any case, I quite liked the idea of Europe: I'd had some great holidays there.

I responded without conviction that I thought the Common Market was not a good idea for Britain. No doubt that was why I didn't get the job .. (only kidding: there were a thousand reasons for the Civil Service not to employ me).

The left is now all at sea over Europe though; the pristine ideological purity of half a century ago has been shredded.

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If the UK capitalist class was truly united in wishing to stay in the EU, that inconvenient referendum result would have been subverted or reversed well before now.

It's certainly true that almost all economists believe that being part of the customs union, with a say on policy, offers by far the best prospects for the UK economy. It goes with the flow of globalisation, increased market size, lowered transaction costs and just-in-time logistics.

But the UK bourgeoisie has never been quite so committed to that predominantly political project which the EU truly is. The path to ever greater convergence, towards that famed economic and political integration into a 'superstate', is in practice one of transition to a unified European polity under German dominance.

With its global interests - the tepid afterglow of Empire - the British establishment always wanted the economic benefits without the political subordination. That balancing act has become ever more difficult as Germany has become more assertive and other members states more reliant upon it, more subservient to its power one might say.

So apart from the most devoted servants of international finance capital in the City (the George Osborne wing of the Tory Party; the Blair wing of Labour) there is a kind of strategic paralysis gripping most other sectors of the capitalist class.

So-called soft-Brexit, which is essentially the status quo ante but with even less political influence in Europe (or, even more disengagement, to put the positive spin on it), is thus the favoured option.

Smaller, more national and regional businesses get little of benefit from the EU and lack lobbying power there. These are the stalwarts of the Tory Party in the country and the most enthusiastic Brexiteers.

Meanwhile everyone is now familiar with the plight of the traditional, relatively unskilled working class, marginalised by higher-skilled, younger and more energetic competition from Eastern Europe. As they have been pressed back into the reserve army of labour, their resentment has fuelled a visceral 'leave' sentiment: a problem for Labour and a tantalising opportunity for the Tories.

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I don't have any views as to what the revolutionary left ought to be campaigning on here. The theory says that you should base your line on objective political analysis, not the vagaries of sentiment of your base-support. Since the EU is just as much a capitalist/globalisation project now as it was 45 years ago, the orthodox Leninist-Trotskyist position must be to oppose it.

But this implies that there's a viable strategy to supersede capitalism (a United Socialist States of Europe!) which can be meaningfully counterposed - and there is not.

The alternate argument, heard from those bright young revisionists in Labour's Momentum, is that socialism has to build on capitalism - it will be even more global and harmonised etc - so we obviously have to back all movement in that direction under capitalism. This reflects the naive idealism of their middle-class professional base which I myself exhibited (and suppressed) in my interview with the Civil Service.*

I'm not here to give advice to British capitalism on their optimal strategy: they don't need much assistance anyway to converge on 'soft-Brexit' as their preferred way forward.

But any institution which lionises George Osborne, Emmanuel Macron and Angela Merkel is going to find me heading rapidly towards the exit.

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* What's wrong with that argument, I hear you say? Simply this: the purpose of capitalism is always and everywhere the valorisation (the self-expansion) of capital. This single-minded dynamic often cuts across broader human interests and values as you may have noticed.

We don't have to completely roll-over for every project of the haute-bourgeoisie to get the benefits of their development of the productive forces. There's a reason they're called 'the bubble'. With effort they can be constrained and diverted.